We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary differential equations (NODEs). Solving systems with fine temporal and spatial grid scales is an ongoing computational challenge, and closure models are generally difficult to tune. Machine learning approaches have increased the accuracy and efficiency of computational fluid dynamics solvers. In this approach neural networks are used to learn the coarse- to fine-grid map, which can be viewed as subgrid-scale parameterization. We propose a strategy that uses the NODE and partial knowledge to learn the source dynamics at a continuous level. Our method inherits the advantages of NODEs and can be used to parameterize subgrid scales, approximate coupling operators, and improve the efficiency of low-order solvers. Numerical results with the two-scale Lorenz 96 ODE, the convection-diffusion PDE, and the viscous Burgers' PDE are used to illustrate this approach.

翻译：我们提出了一种基于神经常微分方程（NODEs）的新方法，用于学习偏微分方程（PDEs）在通过直线法求解及其在混沌常微分方程表示中的子网格尺度模型。在精细时间与空间网格尺度上求解系统是一项持续的计算挑战，而闭合模型通常难以调优。机器学习方法已提升了计算流体力学求解器的精度与效率。该方法中，神经网络被用于学习粗网格到细网格的映射，可视为子网格尺度参数化。我们提出了一种策略，利用NODE与部分知识在连续层面学习源动力学。该方法继承了NODEs的优势，可用于参数化子网格尺度、近似耦合算子，并提升低阶求解器的效率。通过双尺度Lorenz 96常微分方程、对流扩散偏微分方程及粘性Burgers偏微分方程的数值结果，验证了该方法的有效性。